Rotational power in rotational motion is -

Options

A. `vecomega.vectau`
B. `vecomegaxxvectau`
C. `omegaxxtau`
D. `tauxxomega`

Answer

To find the rotational power in rotational motion, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Power**:
Power is defined as the rate at which work is done. In the context of rotational motion, we need to relate this to torque and angular displacement.

2. **Relate Work Done to Torque**:
The work done (W) in rotational motion can be expressed as:
\[ W = \tau \cdot d\theta \]
where \( \tau \) is the torque and \( d\theta \) is the small angular displacement.

3. **Find the Rate of Work Done**:
To find the power (P), we take the derivative of work with respect to time (t):
\[ P = \frac{dW}{dt} = \frac{d(\tau \cdot d\theta)}{dt} \]

4. **Use the Chain Rule**:
By applying the chain rule, we can express this as:
\[ P = \tau \cdot \frac{d\theta}{dt} \]
Here, \( \frac{d\theta}{dt} \) is the angular velocity (\( \omega \)).

5. **Express Power in Terms of Torque and Angular Velocity**:
Substituting \( \omega \) into the equation gives us:
\[ P = \tau \cdot \omega \]
This equation represents the instantaneous power in rotational motion.

6. **Consider the Direction of Vectors**:
If the torque and angular velocity are not in the same direction, we can express the power as the dot product:
\[ P = \tau \cdot \omega \]
This indicates that the power is the component of torque in the direction of angular velocity.

### Final Result:
The rotational power in rotational motion is given by:
\[ P = \tau \cdot \omega \]